Question

How do you simplify \[{{e}^{3}}.{{e}^{4}}\]?

Answer 1

Hint: An exponent is a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression (e.g., 3 in \[{{2}^{3}}=2\times 2\times 2\]). We have to note the value of an exponent is always positive. The product of powers rule is given by the expression \[{{a}^{m}}.{{a}^{n}}={{a}^{m+n}}\].

Complete step by step answer:
As per the given question, we are provided with an exponential expression. We need to simplify the given exponential expression using exponential properties. And, the given exponential expression is \[{{e}^{3}}.{{e}^{4}}\].
We can use the property of the products of power rule to simplify the given exponential expression. This property allows us to simplify problems where we have a product of the same numbers \[a\] raised to two different powers (\[m\]and \[n\]). That is, we can write the formula of this property as
\[\Rightarrow {{a}^{m}}.{{a}^{n}}={{a}^{m+n}}\]
When m is equal to \[-n\] then we get the value as 1.
Here, in the question, we have \[a=e\] and the two powers are \[m=3\] and \[n=4\]. Since the powers of the exponents are different, we can equate the given expression to
\[\begin{align}
  & \Rightarrow {{e}^{3}}.{{e}^{4}}={{e}^{3+4}} \\
 & \Rightarrow {{e}^{3+4}}={{e}^{7}} \\
\end{align}\] \[(\because 3+4=7)\]
\[\therefore {{e}^{7}}\] is the simplified form of the given expression \[{{e}^{3}}.{{e}^{4}}\].

Note:
In order to solve these types of questions, we need to have enough knowledge on exponents. We need to know about the product of powers property in advance to solve this type of problem. The properties of exponents are used in so many problems, so prior knowledge about them is very important.